A Direct Simulation Monte Carlo fluid dynamics code named FENIX has been employed to study gas flow-through properties of the inductively coupled plasma mass spectrometer (ICP-MS). Simulation data have been tested against the Navier-Stokes and heat equations in order to see if FENIX functions properly. The Navier-Stokes and heat equations have been constructed from simulation data and are compared term by term. This comparison shows that FENIX is able to correctly reproduce fluid dynamics throughout the ICP-MS simulation, with an exception immediately behind the ICP-MS sampler cone, where the continuum criterion for the Navier-Stokes equation is not met. Testing the data produced by Fenix also shows that this DSMC method correctly produces momentum and thermal boundary layer phenomenon as well. FENIX output data produce statistical fluctuations of about 2%. Limitations occur from fitting data near surfaces, incurring a relative error of about 5%, and fitting data to take second derivatives where fluid velocity gradients are steep, introducing a relative error of about 10%.

The cause of unintentional initiations of energetic materials is an important area of study due to the risks that comes with storing energetic materials such as high explosives. The current models used to simulate the process of heating energetic materials by a hot metal fragment do not give reliable predictions. The objective of this thesis is to study the current models in order to get a better understanding of how to improve the accuracy of the simulations. The heat transfer in the fragment and energetic material is modeled using the heat equation and the reaction rates in the chemical decomposition of the energetic material are modeled using Arrhenius equations. This study shows the importance of accurately implementing the contact area and heat transfer coefficient between the fragment and the energetic material. The thermal conductivity has a significantly smaller affect on the initiation time compared to the heat transfer coefficient. Furthermore, the dimensions of the fragment affect the resulting simulations greatly, while the dimensions of the energetic material only does so for sufficiently small dimensions. / Orsaken till oavsiktliga initieringar av energetiska material är ett viktigt studieområde på grund av riskerna som följer med att lagra energiskt material, så som sprängämnen. De nuvarande modellerna som används för att simulera uppvärmningsprocessen av energetiska material med ett hett metallfragment ger inte tillförlitliga förutsägelser. Syftet med denna uppsats är att studera de nuvarande modellerna för att få en bättre förståelse för hur man kan förbättra noggrannheten i simuleringarna. Värmeöverföringen i fragmentet och det energetiska materialet modelleras med hjälp av värmeledningsekvationen och reaktionshastigheterna i den kemiska nedbrytningen av det energetiska materialet modelleras med hjälp av Arrhenius-ekvationer. Denna studie visar vikten av att korrekt implementera kontaktytan och värmeöverföringskoefficient mellan fragmentet och det energetiska materialet. Den termiska konduktiviteten har en betydligt mindre effekt på initieringstiden jämfört med värmeöverförings- koefficienten. Vidare så påverkar fragmentets dimensioner de resulterande simuleringarna i hög grad, medan dimensionerna av det energetiska materialet gör så endast för tillräckligt små dimensioner.

Heat transfer is key to understanding many processes in engineering. At a steel mill heat transfer is absolutely crucial to understanding most of the processes. One such a process is the pre-heating of a freshly relined ladle. The goal of this project was to develop code which could solve the diffusion equation, in an arbitrary three-dimensional geometry, subject to Dirichlet, Robin, Neumann, and certain kinds of non-linear boundary conditions. In order to approximate the solution the code uses a cell centred finite volume methodology. In order to verify the computational correctness of the code it was used on three simple cases where analytic solutions are known, a rarity for three-dimensional boundary value problems. A mathematical model for the heat conduction inside a ladle at Ovako’s site in Hofors was developed. The model was evaluated based on measurements on the outside of the ladle as well as from a temperature probe inside the bottom of the ladle. The model was found to adequately agree with the measured temperature. The code can thus be used to find a more optimal heating regiment of the ladle, possibly reducing emissions. / Värmeöverföring är nyckeln till att förstå många processer inom teknik. På ett stålverk är värmeöverföring helt avgörande för att förstå de flesta av processerna. En sådan process är förvärmning av en nymurad skänk. Målet med detta projekt var att utveckla en kod som kunde lösa diffusionsekvationen i en godtycklig tredimensionell geometri under Dirichlet-, Robin-, Neumann- och vissa typer av icke-linjära randvillkor. För att approximera lösningen använder koden en cellcentrerad finita volymmetodik. För att verifiera kodens beräkningsmässiga korrekthet användes den i tre enkla fall där analytiska lösningar är kända, vilket är en sällsynthet för tredimensionella randvärdesproblem. En matematisk modell för värmeledningen i en skänk vid Ovakos anläggning i Hofors utvecklades. Modellen utvärderades utifrån mätningar på utsidan av skänken samt från ett termoelement inuti botten av skänken. Modellen visade sig stämma väl överens med den uppmätta temperaturen. Koden kan därför användas för att hitta ett mer optimalt uppvärmningsschema för skänken, vilket eventuellt kan minska utsläppen från processen.

Dans la première partie de ce mémoire, on s’attache à l’obtention d’Inégalités de Carleman elliptiques pour des opérateurs d’ordre deux au bord pour des conditions dites de Ventcel. Dans une seconde partie, on démontre une Inégalité adaptée aux multi-interfaces, pour des opérateurs elliptiques d’ordre quelconque, sous la condition classique de sous-ellipticité de Hörmander, ainsi que sous une condition de compatibilité entre les opérateurs sur la multi-interface et l’intérieur, dite de recouvrement. Cette condition généralise la condition de Lopatinskii. Enfin, dans une troisième partie, on s’intéresse à la contrôlabilté de l’équation de la chaleur et la stabilisation faible de l’équation des ondes dans des domaines polygonaux. / In the first part of this thesis, we derive elliptic Carleman estimates for second-order operators with Ventcel boundary conditions. In the second part, we prove a proper estimate near multi-interfaces for elliptic operatorsof any order, under the classical sub-ellipticity condition of Hörmander and under a compatibility condition between the operators in the interior and at the multi-interface, called the covering condition. This condition is a generalization of the well-known Lopatinskii condition. Finally, in the third part, we focus on controllability properties of the heat equation, and stabilization properties of the wave equation for polygonal domains, with mixed boundary conditions.

In this thesis we will investigate how the SBP-SAT finite difference method behave with and without an interface. As model problem, we consider the heat equation with piecewise constant coefficients. The thesis is split in two main parts. In the first part we look at the heat equation in one-dimension, and in the second part we expand the problem to a two-dimensional domain. We show how the SAT-parameters are chosen such that the scheme is dual consistent and stable. Then, we perform numerical experiments, now looking at the static case. In the one-dimensional case we see that the second order SBP-SAT method with an interface converge with an order of two, while the second order SBP-SAT method without an interface converge with an order of one.

Hauptthema der Dissertation ist die Analysis eines nichtlinearen, gekoppelten Systems partieller Differentialgleichungen (PDG), das in der Modellierung der Kristallzüchtung aus der Schmelze mit Magnetfeldern vorkommt. Die zu beschreibenden Phenomäne sind einerseits der im elektromagnetisch geheizten Schmelzofen erfolgende Wärmetransport (Wärmeleitung, -konvektion und -strahlung), und andererseits die Bewegung der Halbleiterschmelze unter dem Einfluss der thermischen Konvektion und der angewendeten elektromagnetischen Kräfte. Das Modell besteht aus den Navier-Stokeschen Gleichungen für eine inkompressible Newtonsche Flüssigkeit, aus der Wärmeleitungsgleichung und aus der elektrotechnischen Näherung des Maxwellschen Systems. Wir erörtern die schwache Formulierung dieses PDG Systems, und wir stellen ein Anfang-Randwertproblem auf, das die Komplexität der Anwendung widerspiegelt. Die Hauptfrage unserer Untersuchung ist die Wohlgestelltheit dieses Problems, sowohl im stationären als auch im zeitabhängigen Fall. Wir zeigen die Existenz schwacher Lösungen in geometrischen Situationen, in welchen unstetige Materialeigenschaften und nichtglatte Trennfläche auftreten dürfen, und für allgemeine Daten. In der Lösung zum zeitabhängigen Problem tritt ein Defektmaß auf, das ausser der Flüssigkeit im Rand der elektrisch leitenden Materialien konzentriert bleibt. Da eine globale Abschätzung der im Strahlungshohlraum ausgestrahlten Wärme auch fehlt, rührt ein Teil dieses Defektmaßes von der nichtlokalen Strahlung her. Die Eindeutigkeit der schwachen Lösung erhalten wir nur unter verstärkten Annahmen: die Kleinheit der gegebenen elektrischen Leistung im stationären Fall, und die Regularität der Lösung im zeitabhängigen Fall. Regularitätseigenschaften wie die Beschränktheit der Temperatur werden, wenn auch nur in vereinfachten Situationen, hergeleitet: glatte Materialtrennfläche und Temperaturunabhängige Koeffiziente im Fall einer stationären Analysis, und entkoppeltes, zeitharmonisches Maxwell für das transiente Problem. / The present PhD thesis is devoted to the analysis of a coupled system of nonlinear partial differential equations (PDE), that arises in the modeling of crystal growth from the melt in magnetic fields. The phenomena described by the model are mainly the heat-transfer processes (by conduction, convection and radiation) taking place in a high-temperatures furnace heated electromagnetically, and the motion of a semiconducting melted material subject to buoyancy and applied electromagnetic forces. The model consists of the Navier-Stokes equations for a newtonian incompressible liquid, coupled to the heat equation and the low-frequency approximation of Maxwell''s equations. We propose a mathematical setting for this PDE system, we derive its weak formulation, and we formulate an (initial) boundary value problem that in the mean reflects the complexity of the real-life application. The well-posedness of this (initial) boundary value problem is the mainmatter of the investigation. We prove the existence of weak solutions allowing for general geometrical situations (discontinuous coefficients, nonsmooth material interfaces) and data, the most important requirement being only that the injected electrical power remains finite. For the time-dependent problem, a defect measure appears in the solution, which apart from the fluid remains concentrated in the boundary of the electrical conductors. In the absence of a global estimate on the radiation emitted in the cavity, a part of the defect measure is due to the nonlocal radiation effects. The uniqueness of the weak solution is obtained only under reinforced assumptions: smallness of the input power in the stationary case, and regularity of the solution in the time-dependent case. Regularity properties, such as the boundedness of temperature are also derived, but only in simplified settings: smooth interfaces and temperature-independent coefficients in the case of a stationary analysis, and, additionally for the transient problem, decoupled time-harmonic Maxwell.

Linear Evolution Equations (LEE) have been studied extensively over many years. Their extension in the field of fractional calculus have been defined by Dαu(x, t) = Au(x, t), where α is the fractional order and Dα is a generalized differential operator. Two types of generalized differential operators were applied to the LEE in the state-of-the-art, producing the Riemann-Liouville and the Caputo time fractional evolution equations. However the extension of the new Caputo-Fabrizio derivative (CFFD) to these equations has not been developed. This work investigates existing fractional derivative evolution equations and analyze the generalized linear evolution equations with non-singular ker- nel derivative. The well-posedness of the extended CFFD linear evolution equation is demonstrated by proving the existence of a solution, the uniqueness of the existing solu- tion, and finally the continuous dependence of the behavior of the solution on the data and parameters. Extended evolution equations with CFFD are applied to kinetics, heat diffusion and dispersion of shallow water waves using MATLAB simulation software for validation purpose. / Mathematical Science / M Sc. (Applied Mathematics)

This PhD thesis treats some problems concerning nonlinear differential equations. In the first two papers computer-assisted proofs are used. The differential equations there are rewritten as fixed point problems, and the existence of solutions are proved. The problem in the first paper is one-dimensional; with one boundary condition given by an integral. The problem in the second paper is three-dimensional, and Dirichlet boundary conditions are used. Both problems have their origins in fluid dynamics. Paper III describes an inverse problem for the heat equation. Given the solution, a solution dependent diffusion coefficient is estimated by intervals at a finite set of points. The method includes the construction of set-valued level curves and two-dimensional splines. In paper IV we prove that there exists a unique, globally attracting fixed point for a differential equation system. The differential equation system arises as the number of peers in a peer-to-peer network, which is described by a suitably scaled Markov chain, goes to infinity. In the proof linearization and Dulac's criterion are used.

While moment matching can effectively reduce the dimension of a linear, time-invariant system, it can simultaneously fail to improve the stable time-step for the forward Euler scheme. In the context of a semi-discrete heat equation with spatially smooth forcing, the high frequency modes are virtually insignificant. Eliminating such modes dramatically improves the stable time-step without sacrificing output accuracy. This is accomplished by modal filtration, whose computational cost is relatively palatable when applied following an initial reduction stage by moment matching. A bound on the norm of the difference between the transfer functions of the moment-matched system and its modally-filtered counterpart yields an intelligent choice for the mode of truncation. The dual-stage algorithm disappoints in the context of highly nonnormal semi-discrete convection-diffusion equations. There, moment matching can be ineffective in dimension reduction, precluding a cost-effective modal filtering step.

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